Giant biquadratic interaction-induced magnetic anisotropy in the iron-based superconductor $A_x\mathrm{F}{\mathrm{e}}_{2-y}\mathrm{S}{\mathrm{e}}_2$

The emergence of the electron-pocket-only iron-based superconductor $A_x\mathrm{F}{\mathrm{e}}_{2-y}\mathrm{S}{\mathrm{e}}_2 \left(A=\mathrm{alkali}\mathrm{metal}\right)$ challenges the Fermi-surface nesting picture established in iron pnictides. It was widely believed that magnetism is correlated with the superconductivity in $A_x\mathrm{F}{\mathrm{e}}_{2-y}\mathrm{S}{\mathrm{e}}_2$. Unfortunately, the highly anisotropic exchange parameters and the disagreement between theoretical calculations and experimental results triggered a fierce debate on the nature of magnetism in $A_x\mathrm{F}{\mathrm{e}}_{2-y}\mathrm{S}{\mathrm{e}}_2$. Here we find that the strong magnetic anisotropy is from the anisotropic biquadratic interaction. In order to accurately obtain the magnetic interaction parameters, we propose a universal method, which does not need to include other high-energy configurations as in the conventional energy-mapping method. We show that our model successfully captures the magnetic interactions in $A_x\mathrm{F}{\mathrm{e}}_{2-y}\mathrm{S}{\mathrm{e}}_2$ and correctly predicts the spin-wave spectrum, in quantitative agreement with the experimental observation. These results suggest that the local moment picture, including the biquadratic term, can describe accurately the magnetic properties and spin excitations in $A_x\mathrm{F}{\mathrm{e}}_{2-y}\mathrm{S}{\mathrm{e}}_2$, which sheds new light on the future study of the high-$T_{\mathrm{c}}$ iron-based superconductors.

Figure 1

Magnetic structure for semiconducting $\mathrm{KF}{\mathrm{e}}_{1.5}\mathrm{S}{\mathrm{e}}_2$. Schematic of (a) the stripe AFM order and (b) the order of iron vacancy in $\mathrm{KF}{\mathrm{e}}_{1.5}\mathrm{S}{\mathrm{e}}_2$. Here we assume the direction of antiparallel spin alignment as $\widehat{x}$ and that of parallel spin alignment as $\widehat{y}$.

Figure 2

The spin-wave spectrum calculated by the exchange parameters obtained by (a) our method and (b) the experimental fitting of $\mathrm{KF}{\mathrm{e}}_{1.5}\mathrm{S}{\mathrm{e}}_2$. The black dashed line (red solid line) denotes the spin wave without (with) spin anisotropy. The spin dynamical structure factor (SDSF) of $\mathrm{KF}{\mathrm{e}}_{1.5}\mathrm{S}{\mathrm{e}}_2$ calculated with the exchange parameters obtained by (c) our present method and (d) the experimentally fitted parameters. The intensity of the SDSF has been rescaled to see the optical modes more clearly. Note that the spin-wave spectrum is calculated in the magnetic unit cell whereas the SDSF is calculated in the atomic unit cell.

Figure 3

(a) Schematic of the block-antiferromagnetic order of $\mathrm{KF}{\mathrm{e}}_{1.6}\mathrm{S}{\mathrm{e}}_2$. The square enclosed by the dashed lines denotes the atomic unit cell. (b) The spin-wave spectrum calculated with the exchange parameters from the present method (red line) and the experimental fit results (blue line). Here we assume $S=2$. The inset: the Brillouin zone of the magnetic unit cell in $\mathrm{KF}{\mathrm{e}}_{1.6}\mathrm{S}{\mathrm{e}}_2$.